By using the properties of definite integrals, evaluate the integrals`int_0^ (pi/4) log(1+tanx)dx`

Explanation

To evaluate the integral \( I = \int_0^{\frac{\pi}{4}} \log(1 + \tan x) \, dx \) using the properties of definite integrals, we can follow these steps:

### Step 1: Set up the integral
Let \( I = \int_0^{\frac{\pi}{4}} \log(1 + \tan x) \, dx \).

### Step 2: Use the substitution \( x = \frac{\pi}{4} - t \)
Now, we will change the variable in the integral. Let \( t = \frac{\pi}{4} - x \). Then, \( dx = -dt \) and when \( x = 0 \), \( t = \frac{\pi}{4} \) and when \( x = \frac{\pi}{4} \), \( t = 0 \). Thus, we can rewrite the integral:

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